The query concerning the Euler-Poincare formula’s generalizations - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:54:45Z http://mathoverflow.net/feeds/question/98358 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/98358/the-query-concerning-the-euler-poincare-formulas-generalizations The query concerning the Euler-Poincare formula’s generalizations Jorma Kyppö 2012-05-30T10:58:01Z 2012-05-30T20:29:30Z <p>Euler's equation for polyhedral, Euler's polyhedral formula, V – E + F = 2, where V, E, and F, are the number of points, edges and faces, was discovered by Leonhard Euler in 1752. However, the basic idea - not the equation - was revealed much earlier by Descartes &amp; others, and later generalized by Lhuilier, as follow: V – E + F = 2 – 2g, where g is genus, the number of holes or handles. Later on, Schläfli and Poincare also generalized the formula to the higher dimensional n-polytopes. We talk about Euler-Poincare formula and Euler-Poincare characteristic (X) for combinatorial cell complexes or polyhedral solids: X = N1 – N2 + N3 – N4 + … +/- Nk, where k is the dimension of the complex Nk and X = 2, if k is odd, or 0, if k is even.</p> <p>My <strong>question</strong> is: Are there any other/later generalizations of this Euler-Poincare characteristic of a cell complex?</p> http://mathoverflow.net/questions/98358/the-query-concerning-the-euler-poincare-formulas-generalizations/98370#98370 Answer by Joseph O'Rourke for The query concerning the Euler-Poincare formula’s generalizations Joseph O'Rourke 2012-05-30T12:21:58Z 2012-05-30T12:21:58Z <p>The Wikipedia article discusses (and provides some references for) <a href="http://en.wikipedia.org/wiki/Euler_characteristic#Generalizations" rel="nofollow">several generalizations</a>:</p> <ul> <li>To a <em>chain complex</em>, when the Euler characteristic is the alternating sum of the ranks of the homology groups of the chain complex.</li> <li>To a <em>sheaf</em> on a projective scheme.</li> <li>To an <em>orbifold</em>, which may have a fractional Euler characteristic.</li> <li>To a bounded finite <em>poset</em>.</li> <li>To a finite group or <em>monoid</em>.</li> </ul> http://mathoverflow.net/questions/98358/the-query-concerning-the-euler-poincare-formulas-generalizations/98414#98414 Answer by Jorma Kyppö for The query concerning the Euler-Poincare formula’s generalizations Jorma Kyppö 2012-05-30T20:29:30Z 2012-05-30T20:29:30Z <p>Thank you for your answer. However, I did already study this interesting Wikipedia article. Probably I didn't focus my question enough (first time in mathoverflow). I'm mostly interested about the generalizations, where also the value of genus is aproximated.</p>